I'm trying to implement the exponential fitting scheme in a function like sigma(x)u'' + mu(x)u' + b(x)u = f(x) with the related 2 boundary conditions (page 127 of your book), in order to better understand this scheme.

I figured out an example: u'' + (16x-8)u' - 64xu = 0, that has as exact solution u=x*exp(4x). My x's interval is (0,1) and the boundary conditions are given by the exact solution.

My issue is that the error I get using the scheme sigma_j*D+D-Uj + mu_j*D0Uj + b_j*Uj = f_j is smaller than the error of the scheme rho_j*D+D-Uj + mu_j*D0Uj + b_j*Uj = f_j (with rho_j defined as in your book).

My guess on this strange result (at least form me) is that I'm not using an appropriate function, since I've read on the internet that mu(x)>=U(x_min)>0 (here).

Is my guess correct? can you give me please an example where I can apply this scheme?